We investigate in this paper the complementation of copies of $c_0(I)$ in some classes of Banach spaces (in the class of weakly compactly generated (WCG) Banach spaces, in the larger class $\mathcal{V}$ of Banach spaces which are subspaces of some $C(K)$ space with $K$ a Valdivia compact, and in the Banach spaces $C([1, \alpha ])$, where $\alpha$ is an ordinal) and the embedding of $c_0(I)$ in the elements of the class $\mathcal{C}$ of complemented subspaces of $C(K)$ spaces. Two of our results are as follows:(i) in a Banach space $X \in \mathcal{V}$ every copy of $c_0(I)$ with $\# I < \aleph _{\omega}$ is complemented;(ii) if $\alpha _0 = \aleph _0$, $\alpha _{n+1} = 2^{\alpha _n}$, $n \geq 0$, and $\alpha = \sup \{\alpha _n : n \geq 0\}$ there exists a WCG Banach space with an uncomplemented copy of $c_0(\alpha )$.So, under the generalized continuum hypothesis (GCH), $\aleph _{\omega}$ is the greatest cardinal $\tau$ such that every copy of $c_0(I)$ with $\# I < \tau$ is complemented in the class $\mathcal{V}$. If $T : c_0(I) \to C([1,\alpha ])$ is an isomorphism into its image, we prove that:(i) $c_0(I)$ is complemented, whenever $\| T \| ,\| T^{-1} \| < (3/2)^{\frac 12}$;(ii) there is a finite partition $\{I_1, \dots , I_k\}$ of $I$ such that each copy $T(c_0(I_k))$ is complemented.Concerning the class $\mathcal{C}$, we prove that an already known property of $C(K)$ spaces is still true for this class, namely, if $X \in \mathcal{C}$, the following are equivalent:(i) there is a weakly compact subset $W \subset X$ with ${\rm Dens}(W) = \tau$;(ii) $c_0(\tau )$ is isomorphically embedded into $X$.This yields a new characterization of a class of injective Banach spaces.2000 Mathematical Subject Classification: 46B20, 46B26.

Let X be a Banach space, u ∈ X * * and K, Z two subsets of X * *. Denote by d(u, Z) and d(K, Z) the distances to Z from the point u and from the subset K respectively. The Krein-Šmulian Theorem asserts that the closed convex hull of a weakly compact subset of a Banach space is weakly compact; in other words, every w *-compact subset K ⊂ X * * such that d(K, X) = 0 satisfies d(co w * (K), X) = 0. We extend this result in the following way: if Z ⊂ X is a closed subspace of X and K ⊂ X * * is a w * −compact subset of X * * , then d(co w * (K), Z) ≤ 5d(K, Z). Moreover, if Z ∩ K is w *-dense in K, then d(co w * (K), Z) ≤ 2d(K, Z). However, the equality d(K, X) = d(co w * (K), X) holds in many cases, for instance, if 1 ⊆ X * , if X has w *-angelic dual unit ball (for example, if X is WCG or WLD), if X = 1 (I), if K is fragmented by the norm of X * * , etc. We also construct under CH a w *-compact subset K ⊂ B(X * *) such that K ∩ X is w *-dense in K, d(K, X) = 1 2 and d(co w * (K), X) = 1.

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Abstract. We introduce and study the Kunen-Shelah properties KS i , i = 0, 1, . . . , 7. Let us highlight some of our results for a Banach space X: (1)

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